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Prelude Duality A Final V -coalgebra Future Work A final Vietoris coalgebra beyond compact spaces and a generalized J onsson-Tarski duality Liang-Ting Chen School of Computer Science, University of Birmingham 26 July 2011

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Prelude Duality A Final V-coalgebra Future Work

A final Vietoris coalgebra beyond compact spaces and a generalized J´

  • nsson-Tarski duality

Liang-Ting Chen

School of Computer Science, University of Birmingham

26 July 2011

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J´

  • nsson-Tarski duality is . . .

1 in old days . . .

MA(BA)

  • U
  • DGF
  • U
  • BA

Ultra Stone Clop

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J´

  • nsson-Tarski duality is . . .

1 in old days . . .

MA(BA)

  • U
  • DGF
  • U
  • BA

Ultra Stone Clop

  • 2 nowadays . . .

Alg(M)

  • U
  • Coalg(V)
  • U
  • BA

Pt Idl Stone KΩ

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Is is true in general?

The main purpose of this talk: Alg(M)

  • Coalg(V)
  • Frm

Pt

Top

Ω

  • and an application, the final V-coalgebra.

Let’s see how far we can go.

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Kripke frames are P-coalgebras

Definition A Kripke frame X, R consists of

1 a set X and 2 a relation R of X

ξR :X → PX x → {y : xRy}

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Bounded morphisms are P-coalgebra morphisms

Definition A bounded morphism f : X, R → Y , S is a functional bisimulation. X

ξ

  • f
  • PX

Pf

  • Y

η PY

That is, f is a P-coalgebra morphism.

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Descriptive general frames are V-coalgebras

Definition A descriptive general frame is

1 a Kripke frame X, ξ : X → PX 2 on a Stone space X, B

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Descriptive general frames are V-coalgebras

Definition A descriptive general frame is

1 a Kripke frame X, ξ : X → PX 2 on a Stone space X, B 3 ξ(x) ∈ KX 4 B is closed under 1

V = {x : ξ(x) ⊆ V }

2

♦V = {x : ξ(x) ∩ V = ∅}.

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Descriptive general frames are V-coalgebras

Definition A descriptive general frame is

1 a Kripke frame X, ξ : X → PX 2 on a Stone space X, B 3 ξ(x) ∈ KX 4 B is closed under 1

V = {x : ξ(x) ⊆ V }

2

♦V = {x : ξ(x) ∩ V = ∅}.

ξi : X, Bi → KX, ? What is the finest topology making every ξi continuous?

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Definition A Vietoris topology VX = KX, τ of a Stone space X is a Stone space with τ generated by

1 V = {K ∈ KX : K ⊆ V } 2 ♦V = {K ∈ KX : K ∩ V = ∅} where V ∈ KΩX.

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Definition A Vietoris topology VX = KX, τ of a Stone space X is a Stone space with τ generated by

1 V = {K ∈ KX : K ⊆ V } 2 ♦V = {K ∈ KX : K ∩ V = ∅} where V ∈ KΩX.

KΩX

  • ΩVX

ξ−1 ΩX

ξ−1(V ) = {x : ξ(x) ∈ V } = {x : ξ(x) ⊆ V } similarly for ξ−1(♦V ).

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Bounded morphisms are V-coalgebra morphisms

Definition A morphism f : X, R, B → Y , S, C is a bounded morphism and also f −1(C) ∈ B for any C ∈ C (continuity). Fact Vf (K) = f [K] is compact Vf is continuous.

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Bounded morphisms are V-coalgebra morphisms

Definition A morphism f : X, R, B → Y , S, C is a bounded morphism and also f −1(C) ∈ B for any C ∈ C (continuity). Fact Vf (K) = f [K] is compact Vf is continuous. X

ξ

  • f
  • PX

Pf

  • Y

η PY

X

ξ

  • f
  • VX

Vf

  • Y

η

VY

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Modal algebras are M-algebras

Definition Modal algebra = Boolean algebra + unary operators , ♦ subject to normal modal logic laws

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Modal algebras are M-algebras

Definition Modal algebra = Boolean algebra + unary operators , ♦ subject to normal modal logic laws Definition A modal algebra construction MA of a Boolean algebra is BA♦A ∪ A | (a ∧ b) = a ∧ b ♦(a ∨ b) = ♦a ∨ ♦b ♦(a ∧ b) ≥ ♦a ∧ b (a ∨ b) ≤ ♦a ∨ b

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A M-algebra is an interpretation: MA

α

A → A, , ♦

a = α(a), ♦a = α(♦a).

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A M-algebra is an interpretation: MA

α

A → A, , ♦

a = α(a), ♦a = α(♦a). Conversely, we can obtain an interpretation by freeness: MA

  • id

A

A, , ♦

a→a,♦a→♦a

  • id
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Modal algebra morphisms are M-algebra morphisms

Definition A modal algebra morphism f is a Boolean algebra morphism and f (Aa) = Bf (a), f (♦Aa) = ♦Bf (a) and relations. A

A f

  • A

f

  • B

B

B

Modal algebra morphisms are M-algebras morphisms.

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Definition Given f : A → B, define

1 Mf (a) = f (a) and Mf (♦a) = ♦f (a) 2 Mf is a Boolean algebra homomorphism by freeness of MA.

MA

α

  • Mf
  • A
  • f
  • MB

β

B

f (Aa) = (f ◦ α)(a) = β(f (a)) = Bf (a)

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Short summary

Fact

1 DGF ∼

= Coalg(V) and MA(BA) ∼ = Alg(M)

2 The classical J´

  • nsson-Tarski duality: DGF ∼

= MA(BA)op

3 The (co)-algebra viewpoint: Coalg(V) ∼

= Alg(M)op

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Short summary

Fact

1 DGF ∼

= Coalg(V) and MA(BA) ∼ = Alg(M)

2 The classical J´

  • nsson-Tarski duality: DGF ∼

= MA(BA)op

3 The (co)-algebra viewpoint: Coalg(V) ∼

= Alg(M)op Question

1 What is the relationship between M and V? 2 An extension of M and V?

BA

M

  • Idl
  • BA

Idl

  • Frm

MF

Frm

Stone

V

  • J
  • Stone

J

  • Top

V′

Top

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Duality between algebras and coalgebras

Fact Coalg(T)op ≡ Alg(T op) where T op : X op → X op x → x f op → (Tf )op

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Dual functors

T is dual to L if X op

T op

  • F
  • A

L

  • X op
  • F

A i.e. LF ∼ = FT. Fact Coalg(T)op ∼ = Alg(T op) ∼ = Alg(L) if X op ∼ = A and LF ∼ = FT.

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Topological spaces and Frames

A dual adjunction . . . Frm

Pt Top Ω

  • ΩX = the complete lattice of open sets with
  • S ∧ a =
  • s∈S

(s ∧ a)

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Topological spaces and Frames

A dual adjunction . . . Frm

Pt Top Ω

  • ΩX = the complete lattice of open sets with
  • S ∧ a =
  • s∈S

(s ∧ a) Pt A = the space of frame homomorphisms f : A → 2 with open sets Ua = {ϕ ∈ Pt A : ϕ(a) = ⊤}

  • r, neighbourhoods systems.
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Sober spaces and spatial frames

Definition A frame A is spatial if the unit is an iso. A space X is sober if the unit is an iso.

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Sober spaces and spatial frames

Definition A frame A is spatial if the unit is an iso. A space X is sober if the unit is an iso. A cheat dual equivalence . . . SFrm

Pt

Sob

Ω

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Definition A modal algebra construction MFA of a frame A is FrmA ∪ ♦A| normal modal logic laws with the following for any directed S ⊆ A

1 ( S) =

s∈S s

2 ♦( S) =

s∈S ♦s

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Definition A modal algebra construction MFA of a frame A is FrmA ∪ ♦A| normal modal logic laws with the following for any directed S ⊆ A

1 ( S) =

s∈S s

2 ♦( S) =

s∈S ♦s

BA

M

  • Idl
  • BA

Idl

  • Frm

MF

Frm

MF is an extension of M along Idl.

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Stably locally compact frames

Unfortunately, SFrm is not closed under MF. Stably locally compact frames (locales) are closed under MF, i.e. MF : SLCFrm → SLCFrm. Definition A stably locally compact frame is

1 a continuous domain 2 x ≪ y1 and x ≪ y2 ⇒ x ≪ y1 ∧ y2

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Stably locally compact frames

Unfortunately, SFrm is not closed under MF. Stably locally compact frames (locales) are closed under MF, i.e. MF : SLCFrm → SLCFrm. Definition A stably locally compact frame is

1 a continuous domain 2 x ≪ y1 and x ≪ y2 ⇒ x ≪ y1 ∧ y2

Example

1 the ideal completion of Boolean algebras 2 the ideal completion of distributive lattices 3 compact regular frames 4 . . .

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Stably locally compact spaces

By dual equivalence, SLCFrm ∼ = SLCSpop: Definition A space X is stably locally compact if X ∈ Sob and ΩX ∈ SLCFrm.

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Stably locally compact spaces

By dual equivalence, SLCFrm ∼ = SLCSpop: Definition A space X is stably locally compact if X ∈ Sob and ΩX ∈ SLCFrm. Example

1 Stone spaces 2 coherent spaces, i.e. Priestly spaces 3 compact Hausdorff spaces 4 . . .

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Is MF dual to any functor?

Definition V′ : Top → Top V′X = KLX, τ where

1 LX = the set of intersections of open and closed sets 2 τ = U ∨ ♦U

U = {L : L ⊆ U}, ♦U = {L : L ∩ U = ∅} where U ∈ ΩX. SLCFrm

MF

  • Pt

SLCSp

V′

  • SLCFrm

Pt

SLCSp

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A generalized modal duality

Finally . . . Alg(MF)

  • Coalg(V′)
  • SLCFrm

Pt

SLCSp

Ω

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A canonical Kripke structure

Fact A descriptive general Kripke frame X, R, B is canonical if and

  • nly if ξR : X → VX is a final V-coalgebra.
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A canonical Kripke structure

Fact A descriptive general Kripke frame X, R, B is canonical if and

  • nly if ξR : X → VX is a final V-coalgebra.

Now, turn to our setting: How to find a V′-coalgebra? Fact A final V′-coalgebra corresponds to an initial MF-algebra.

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MF-initial sequence

Base case (the initial object in Frm): 2

  • . . .

Inductive case: → Mn

F2 → Mn+1 F

2 → and transfinitely by colimits.

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MF-initial sequence

Base case (the initial object in Frm): 2

  • . . .

Inductive case: → Mn

F2 → Mn+1 F

2 → and transfinitely by colimits. Fact If the sequence converges, i.e. α : MF κ2 ∼ = MFMκ

F2 for some κ,

then α−1 : MFMκ

F2 → Mκ F2

is an initial MF-algebra.

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is infinitary, so it is not clear whether MF converges. Fact

1 2 = Idl2 2 MF preserves coherences, i.e.

DLat

MD Idl

  • DLat

Idl

  • Frm

MF

Frm

3 MD is the same as MF but it is a free distributive

construction.

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is infinitary, so it is not clear whether MF converges. Fact

1 2 = Idl2 2 MF preserves coherences, i.e.

DLat

MD Idl

  • DLat

Idl

  • Frm

MF

Frm

3 MD is the same as MF but it is a free distributive

construction.

4 DLat is finitary, so MD converges at ω.

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2

  • MD2

M2

D2

. . . Mω

D2 ∼

= MDMω

D2

Mω

D is a colimit.

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2

  • MD2

M2

D2

. . . Mω

D2 ∼

= MDMω

D2

Mω

D is a colimit. MF-initial sequence converges at ω:

MFMω

F2 ∼

= MFIdlMω

D2

{Idl preserves colimits} ∼ = IdlMDMω

D2

{MF preserves coherence} ∼ = IdlMω

D2

{MD converges at ω} ∼ = Mω

F2

{Idl preserves colimits}

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Future work

1 V : Stone → Stone can be obtained by Pω via

Pro-completion.

2 MF : Frm → Frm can be obtained by Pω via relation lifting.

See Y. Venema’s Generalized Powerlocales via Relation Lifting.

3

SetSetω

ω R.L. PosetPosetω ω

DLatDLat

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Put things together . . .

1 A big picture:

SetSetω

ω R.L.

  • DLatDLat

c.b?

  • ?
  • Pt Ω∗ PriesPries

?

  • FrmFrm

TopTop

where c.b. is the change of base from Poset to DCPO.

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Put things together . . .

1 A big picture:

SetSetω

ω R.L.

  • DLatDLat

c.b?

  • ?
  • Pt Ω∗ PriesPries

?

  • FrmFrm

TopTop

where c.b. is the change of base from Poset to DCPO.

2 A description of Pt ◦MT ◦ Ω where T : Setω → Setω?

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Thank you for your attention! For details, please see my extended abstract, Vietoris locales by P. Johnstone, and Stone coalgebras by C. Kupke.